分數階變革式奈米Duffing共振器系統的渾沌及其同步與反控制

Abstract

本篇論文以相圖、龐卡萊映射圖及分歧圖等數值方法研究分數階變革式奈米Duffing共振器系統的渾沌行為。基於頻域的觀點，零到一階之間的分數階積分器可以線性轉移函數的近似計算而得。可以發現系統總階數1.8、1.9、2.0、2.1時，系統具有渾沌現象。兩個沒有耦合的分數階變革式奈米Duffing共振器系統之渾沌同步可以藉以第三渾沌系統的渾沌狀態變數之相同函數取代它們相對應的參數而達成。此方法稱為參數激發渾沌同步。渾沌同步可成功地以很低的總分數階數0.2獲得，數值模擬見於相圖、龐卡萊映射圖和狀態誤差圖。最後研究分數階變革式奈米Duffing共振器系統的反控制。首先，以第二個全同的系統之狀態變數的函數作為添加項，渾沌反控制即可獲得。接著以白噪訊、Rayleigh噪訊、Rician噪訊、均勻噪訊等分別作為添加項，渾沌反控制亦可獲得。渾沌反控制可成功地以很低的總分數階數0.2達成。數值模擬以相圖、龐卡萊映射圖表示。

In this thesis, the chaotic behaviors in a fractional order modified nano Duffing resonator system are studied numerically by phase portraits, Poincaré maps and bifurcation diagrams. Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in (0,1], based on frequency domain arguments. The total system orders found for chaos to exist in such systems are 1.8, 1.9, 2.0 and 2.1. The chaos synchronizations of two uncoupled fractional order chaotic modified nano Duffing resonator systems are obtained. By replacing their corresponding parameters by the same function of chaotic state variables of a third chaotic system, the chaos synchronization can be obtained. The method is named parameter excited chaos synchronization which can be successfully obtained for very low total fractional order 0.2. Numerical simulations are illustrated by phase portrait, Poincaré map and state error plots. Anti-control of chaos of a fractional order modified nano Duffing resonator system is studied. First, by using the functions of state variable of a second identical system as the added term, the anti-control of chaos can be obtained. Second, by using the white noise, Rayleigh noise, Rician noise and uniform noise as the added term respectively, the anti-control of chaos can be obtained. Anti-control of chaos can be successfully obtained for very low total fractional order 0.2. Numerical simulations are illustrated by phase portraits and Poincaré maps.